ROBUSTNESS OF PARAMETRIC AND NONP ARAMETRIC TESTS UNDER NON-NORMALITY FOR TWO INDEPENDENT SAMPLE 1USMAN.M, 2IBRAHIM. N 1Dept of Statistics, Fed. Polytechnic Bali, Taraba State, Dept of Maths and Statistcs, Fed. Polytechnic MubiNigeria E-mail: [email protected] __________________________________________________________________________________________ Abstract - Robust statistical methods have been developed for many common problems, such as estimating location, scale and regression parameters. One motivation is to provide methods with good performance when there are small departures from parametric distributions. This study was aimed to investigates the performance oft-test, Mann Whitney U test and Kolmogorov Smirnov test procedures on independent samples from unrelated population, under situations where the basic assumptions of parametric are not met for different sample size. Testing hypothesis on equality of means require assumptions to be made about the format of the data to be employed. Sometimes the test may depend on the assumption that a sample comes from a distribution in a particular family; if there is a doubt, then a non-parametric tests like Mann Whitney U test orKolmogorov Smirnov test is employed. Random samples were simulated from Normal, Uniform, Exponential, Beta and Gamma distributions. The three tests procedures were applied on the simulated data sets at various sample sizes (small and moderate) and their Type I error and power of the test were studied in both situations under study. Keywords - Non-normal,Independent Sample, T-test,Mann Whitney U test and Kolmogorov Smirnov test. I. INTRODUCTION the data to be an independent random sample4. In nonparametric tests very few assumptions are made Robust statistics are statistics with good performance about the distribution underlying the data and, in for data drawn from a wide range of probability particular, it is not assumed to be a normal distributions, especially for distributions that are not distribution. Some statisticians prefer to use the term normal. It seeks to provide methods that emulate distribution-free rather than nonparametric to popular statistical methods, but which are not unduly describe these tests5 affected by outliers or other small departures from model assumptions. In statistics, classical estimation Nonparametric statistical tests are concerned with the methods rely heavily on assumptions which are often application of statistical data in nominal or ordinal not met in practice. In particular, it is often assumed scale to problems in pure science, social science, that the data errors are normally distributed, at least engineering and other related fields. Most of the approximately, or that the central limit theorem can present analysis carried out by non science and be relied on to produce normally distributed science oriented researchers are based on parametric estimates1.Nonparametric tests are usually less test, and it is often reasonable to assume that powerful than corresponding tests designed for use on observations come from a particular family of data that come from a specific distribution. Thus, you distributions. Moreover, experience backed by theory, are less likely to reject the null hypothesis when it is suggest that for measurements, inferences based on false. The tests often require one to modify the the assumption that observations form a random hypotheses. For example, most nonparametric tests sample from some normal distribution may not be about the population center are tests about the median misleading even if the normality assumption is instead of the mean. The test does not answer the incorrect, but this is not always true6. These tests same question as the corresponding parametric often are used in conjunction with small samples, procedure2. The strength of a nonparametric test because for such samples the central limit theorem resides in the fact that it can be applied without any cannot be invoked. Nonparametric tests can be assumption on the form of the underlying directed toward hypothesis concerning the form, distribution. It is good for data with outliers and work dispersion or location (median) of the population. In well for ordinal data (data that have a defined order) the majority of the applications, the hypothesis is because it based on ranks of data3.Nonparametric test concerned with the value of a median, the difference is a hypothesis test that does not require the between medians or the differences among several population's distribution to be characterized by medians. This contrasts with the parametric certain parameters. For instance, many hypothesis procedures that are focused principally on population tests rely on the assumption that the population means. If normal model cannot be assumed for the follows a normal distribution with parameters μ and data then the tests of hypothesis on means are not σ. Nonparametric tests do not have this assumption, applicable. Nonparametric tests were created to so they are useful when your data are strongly overcome this difficulty. Nonparametric tests are nonnormal and resistant to transformation. However, often (but not always) based on the use of ranks; such nonparametric tests are not completely free of as Kolmogorov Smirnov test, Wilcoxon rank test, assumptions about your data, but it may also require Proceedings of 105th The IRES International Conference, Putrajaya, Malaysia, 4th-5th March, 2018 1 Robustness of Parametric and Nonp Arametric Tests under Non-Normality for Two Independent Sample Sign test, Mann Whitney U test, Kruskal Wallis test, values and not widely spread out (the standard etc7,8. deviation is low)9.Note that before performing any In this article some hypothesis tests were developed Independent t-Test the assumptions of independency, in situations where the data come from a probability normality and homogeneity of variance must be distribution whose underlying distribution is normal satisfied. or not and different sample size were considered for Let x1iandx2i be two of observation from different each case of an independent sample. If the population of individual i, then we obtain observations from two samples are unrelated, then we x 1 − x 2 t = ~ tn +n −1 (1) have independent observations. For example, a test 1 1 1 2 Sp + could be done to investigate whether there is a n1 n2 difference in test anxiety based on educational level st nd where x 1 is the 1 sample mean, x 2 is the 2 sample (i.e., your dependent variable would be "test anxiety" 2 2 2 n1−1 S1+ n2−1 S2 mean andSp = The null hypothesis and your independent variable would be "educational n1+n2−2 level", which has two groups: "undergraduates" and (H0) is reject if calculated t value is greater than table α "postgraduates"). Unrelated samples, also called value (i.e. t cal > t ,n +n −2) otherwise we do not 2 1 2 unpaired samples or independent samples, are 9 reject null hypothesis . samples in which the cases in each sample are different. Often we are investigating differences in 2.2 Mann Whitney U test individuals, which means that when comparing two Mann Whitney U-test is a nonparametric test samples, an individual in one sample cannot also be a alternative to the two independent sample t-test, member of the other sample and vice versa. An which is called the Wilcoxon rank sum test. The test, example would be gender, an individual would have sometimes called the Mann Whitney U Test or the to be classified as either male or female not both. Wilcoxon Rank Sum Test, is used to test whether two The objectives of the studywas aimed on examining samples are likely to derive from the same population the effect of non-normality on parametric (i.et-test) (i.e., that the two populations have the same shape). and the nonparametric tests (i.e. Mann Whitney U test Some investigators interpret this test as comparing and Kolmogorov Smirnov test)effect. Also to the medians between the two populations. The examine the effects of sample size on the three test procedure for the test involves pooling the procedures based on type I error and power of test. observations from the two samples into one combined sample and arrange the jointly values in an increasing II. MATERIALS AND METHODS order of magnitude and rank them in ascending order of magnitude, keeping track of which sample each The materials used for the analysis were generated observation comes from, [i.e. from 1 to (n + n )], data using simulation procedures fromselected 1 2 respectively. If there is ties, we assigned to each of continuous distributions. Since it is very difficult to the tied observations the mean of the ranks which get data that follows these distribution patterns, even they occupy. The test is based on W the sum of the if there is, it is very difficult to get the required 1 ranks of the values of first group or on W the sum of number of replicates for the sample sizes of interest. 2 the ranks of the values of second group. In actual The parametric (Independent t-test) and practice, we base the test directly on W and W , nonparametric (Mann Whitney U test and 1 2 instead we use the related statistics. Kolmogorov Smirnov test), methods of analyzing n1 n1 + 1 independent sample were applied, to compare the U1 = W1 − or performance of each test on the generated data from 2 n n + 1 the Normal, Uniform, Exponential, Beta and Gamma 2 2 U2 = W2 − (2) distributions based on the underlying criteria for 2 or the smaller of the two, which we denote by U i.e. assessment. U = min(U1, U2). This test is often performed as a two-sided test and, thus, the hypothesis indicates that 2.1 Student’s Independent t-test the populations are not equal as opposed to specifying The independent t-test, also called the two sample t- directionality. A one-sided hypothesis is used if test or student's t-test, is an inferential statistical test interest lies in detecting a positive or negative shift in that determines whether there is a statistically one population as compared to the other.Therefore we significant difference between the means in two unrelated samples on the same continuous, dependent reject H0 when U ≤ Uα 2 for two tailed test or when U2 ≤ Uα for one tailed test.